Abstract
The aim of this paper is to investigate the asymptotic behavior of positive solutions to the following degenerate and singular parabolic system
where the constants \({0 \leq \alpha, \beta < 1, p_{1}, p_{2}, q_{1}, q_{2}, k_{1}, k_{2} > 0}\). Under appropriate hypotheses, we first prove a local existence of classical solution by a regularization method. Then, we discuss the global existence and blowup of positive solutions by using a comparison principle. Finally, we give the precise blowup rate estimates and the uniform blowup profiles by using the method developed by Souplet [19].
Similar content being viewed by others
References
Aronson D., Crandall M.G., Peletier L.A.: Stabilization of a degenerate nonlinear diffusion problem. Nonlinear Anal. 6, 1001–1022 (1982)
Bebernes J., Bressan A., Lacey A.: Total blow-up versus single point blow up. J. Differ. Equ. 73, 30–44 (1988)
Chadam J., Peirce A., Yin H.M.: The property of solutions to some diffusion equation with localized nonlinear reactions. J. Math. Anal. Appl. 169, 313–328 (1992)
Chan C.Y., Chen C.S.: A numerical method for semilinear singular parabolic quenching problem. Quart. Appl. Math. 47, 45–57 (1989)
Chen Y.P., Liu Q.L., Xie C.H.: The blow-up properties for a degenerate semilinear parabolic equation with nonlocal source. Appl. Math. J. Chin. Univ. Ser. B 17, 413–424 (2002)
Chen Y.P., Xie C.H.: Blow-up for degenerate, singular, semilinear parabolic equations with nonlocal source. Acta Math. Sinica 47, 41–50 (2004)
Dunford N., Schwartz J.T.: Linear Operators, Part 2: Spectral Theory, Self-Adjoint Operatorsin Hilbert Space. Interscience, New York (1963)
Friedman A., Mcleod B.: Blow-up of positive solutions of semilinear heat equations. India. Univ. Math. J. 34, 425–447 (1985)
Galationov V.A., Levine H.A.: A general approach to critical Fujita exponents in nonlinear parabolic problems. Nonlinear Anal. 34, 1005–1027 (1998)
Giga Y., Kohn R.V.: Asymptotic self-similar blow-up of semilinear heat equations. Comm. Pure Appl. Math. 38, 297–319 (1985)
Ladde G.S., Lakshmikantham V., Vatsala A.S.: Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman, Boston (1985)
Ladyzenskaja O.A., Solonik V.A., Ural’ceva N.N.: Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, vol 23. American Mathematical Society, Rhode Island (1967)
Li F.C., Huang S.X., Xie C.H.: Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discret. Contin. Dyn. Syst. 9, 1519–1532 (2003)
Liu Q.L., Chen Y.P., Xie C.H.: Blow-up for a degenerate parabolic equation with nonlocal source. J. Math. Anal. Appl. 285, 487–505 (2003)
Mclachlan N.W.: Bessel Functions for Engineers, 2nd edn. Oxford at the Claredon Press, London (1955)
Pao C.V.: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992)
Peng C.M., Yang Z.D., Xie B.L.: Global existence and blow-up for degenerate and singular nonlinear parabolic system with nonlocal source. Nonlinear Anal. 72, 2474–2487 (2010)
Souplet P.: Blow-up in nonlocal reaction-diffusion equation. SIAM J. Math. Anal. 29, 1301–1334 (1998)
Souplet P.: Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source. J. Differ. Equ. 153, 374–406 (1999)
Wang M.X., Wang Y.M.: Properties of positive solutions for nonlocal reaction-diffusion problems. Math. Methods Appl. Sci. 19, 1141–1156 (1996)
Zhou J., Mu C.L., Li Z.P.: Blow up for degenerate and singular parabolic system with nonlocal source. Bound. Value Probl. 2006, 1–19 (2006)
Zhou J., Mu C.L.: Blowup for a degenerate and singular parabolic equation with nonlocal source and absorption. Glasgow Math. J. 52, 209–225 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by NSFC grant 11201380, the Fundamental Research Funds for the Central Universities grant XDJK2012B007, Doctor Fund of Southwest University grant SWU111021 and Educational Fund of Southwest University grant 2010JY053.
Rights and permissions
About this article
Cite this article
Zhou, J. Global existence and blowup for a degenerate and singular parabolic system with nonlocal source and absorptions. Z. Angew. Math. Phys. 65, 449–469 (2014). https://doi.org/10.1007/s00033-013-0342-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00033-013-0342-0